1. Converting the buffer layer to semiconducting graphene and
the role of incommensurate mutual modulation
A Thesis Proposal
Presented to
The Academic Faculty
by
Matthew Conrad
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
School of Physics
Georgia Institute of Technology
May 2016
2. Converting the buffer layer to semiconducting graphene and
the role of incommensurate mutual modulation
Approved by:
Professor Edward Conrad, Chair
Professor Philip First
Professor Martin Mourigal
4. CHAPTER I
INTRODUCTION
In 2004 graphene[1, 2] captured the attention of the condensed matter community and
claimed the Nobel Prize in 2010. It was one of the first truly 2D materials available for
rigourous study and opened up an entirely new field of research. The discovery of graphene
was surprising as 2D materials were long thought to be unstable and the electron transport in
graphene resembled the behavior of massless Dirac fermions [3]. Even more excitement grew
as graphene showed great promise for applications through remarkable electronic properties
like long range ballistic conduction at room temperature and record mobilities[4].
Recently, progress towards graphene-based electronics has halted due to the inability
to develop a viable form of semiconducting graphene. As a result, research has shifted to
other less favorable materials such as 2D metal dichalcogenides. A promising system for
semiconducting graphene is the first graphene layer grown on SiC(0001) (the buffer layer)
due to its strong and ordered interaction with SiC. However, a proper understanding of the
buffer layer has been elusive because theoretical studies were too computationally expensive
and growth methods inhibited comprehensive experimental studies. However, improved
growth methods are now available for detailed electronic and structural measurements.
Such studies are essential to enable a proper theoretical understanding and to assess if
buffer graphene can be tailored into a viable form of semiconducting graphene.
This thesis proposal outlines a path of study for the buffer layer from an experimen-
tal and theoretical perspective. First, a literature review is presented that outlines what
graphene is and how to it semiconducting. Then a review of graphene on SiC and a history
of the buffer layer will be provided. Following the literature review, detailed measurements
of the first surface X-ray diffraction (SXRD) measurements on the buffer layer are coupled
with angle resolved photomession spectroscopy (ARPES) measurements to correlate the
atomic and electronic structure through a simple tight binding model. Finally, an outline
1
5. a
b
τ1
τ2
τ3
a b
ΓK KM
E−EF(eV)
30
20
10
0
-10
c
0.1
0
-0.11.61.7
1.8
-0.5
0
0.5
kx (˚A−1)ky (˚A−1)
E−EF(eV)
-5
0
5
E−EF(eV)
DOS (a.u)
d
a∗
b∗K
Γ K
M
Figure 1.1: (a) Atomic structure of graphene. The gray diamond represents the unit
cell. “A” (“B”) atoms are in gray (gold). The red arrows represent the lattice vectors.
Black arrows represent the vectors for the three nearest neighbors to an atom A atom. (b)
Band structure of graphene π-bands (yellow) and σ-bands (gray) calculated from the tight
binding method. The two π-bands touch at the K-point(and K ) with a linear dispersion
that give rise to a Dirac cone shown in (c). The insert is a schematic of reciprocal space.
The reciprocal lattice vectors in blue, a∗ and b∗, are determined from the lattice vectors in
(a), a and b, respectively. The boundary of the first Brillioun zone is indicated by the gray
hexagon. The band structure was calculated along the path shown in red. (d) density of
states of graphene pi-bands.
of the future work to conduct detailed experimental and theoretical studies of the buffer
will be presented.
1.1 Graphene
Graphene is a planar two-dimensional hexagonal crystal of carbon atoms [see Fig. 1.1(a)].
The positions of the atoms are described by lattice and basis vectors, R = Rm,n + r. The
lattice positions are determined by integer multiples of the lattice vectors, Rm,n = ma + nb
where the lattice vectors are rotated by 2π/3. The magnitude of a and b are equal and
define the graphene lattice parameter, ag ≈ 2.46 ˚A. The hexagonal structure is formed by
having two equivalent basis atoms, where the “A” atom is located at the origin and the “B”
atom at RB = 1/3 a − b .
The reciprocal lattice of graphene is also hexagonal where a∗ and b∗ are rotated by π/3
and have a magnitude of 4π/ag
√
3 as shown in the insert in 1.1(b). The symmetry points
of interest are Γ = 0, M = 1/2a∗, K± = ±1/3(a∗ + b∗).
2
6. Each carbon atom has four valence electrons. Three of the electrons in the 2s, 2px, and
2py orbitals sp2 hybridize to form strong σ-bonds to the three neighboring carbon atoms.
The remaining electron in the 2pz state delocalizes and forms π-bonds with the neighbor-
ing carbon atoms. In perfectly flat graphene the π-bonds can be considered separately
because they are orthogonal to the σ-bonds. Furthermore, the π-electrons are responsible
for the low energy electronic structure and were first studied theortically using the tight
binding method to serve as a building block for understanding the electronic properties
of graphite[5]. Despite its simplicity, the tight binding (TB) model predicts experimental
results[6] and agrees with more rigourous ab initio calculations[7] . For this reason, TB
models are often the starting point for exploring new phenomena in the electronic structure
of graphene systems.
At low energy, the band structure of the π-electrons have two bands (bonding and
anti-bonding) that touch at the K±-points with linear dispersion and azimuthal symme-
try[See Fig. 1.1(b) and (c)]. The band structure is classified as a zero gap semiconductor
or a semimetal since the density of states is minimum where the two bands touch[See Fig.
1.1(d)]. Although the band dispersion is the same at K±-points, pseudospin arises from the
inequivalence of the wavefunctions at K±-points[See the insert in Fig. 1.1(b)]. The pres-
ence of linear dispersion and pseudospin at low energy indicate that the quasiparticles in
graphene behave like massless Dirac fermions and open new methods for studying quantum
electrodynamics in a condensed matter system[3]. As such, the shape of the low energy
dispersion is characterized by a Dirac cone and the two bands touch at the Dirac point.
The reason for this unique gapless electronic structure in graphene is due to sublattice
symmetry, i.e. both A and B atoms are carbon, and hexagonal symmetry. The combination
of these symmetries ensures a degeneracy in energy at the K±-points[8]. The methods
for producing bandgaps in graphene involve breaking both or either of these symmetries.
Symmetry-breaking methods fall into three broad classes: quantum confinement, strain and
functionalization.
Theoretically, defining graphene into certain shapes of sufficiently small size may in-
duce bandgaps. Two notable examples are nanoribbons[9] and antidots[10]. Both quantum
3
7. confinement and edge termination play important roles in these systems. The effects of
quantum confinement are demonstrated by the band gap being inversely proportional to
the ribbon width or size of the antidot. Differences in edge termination effects the size of
the band gap and the presence of zero energy states. The challenge facing nanoribbons
lies in the necessity to fabricate crystallographically defined edges with minimal defects at
small and precise ribbon widths or antidot size. Uniaxial strain can open substaintial band
gaps in graphene, but require large strains beyond the elastic limit and reduces the Fermi
velocity[11]. Other types of local strain show promise of opening gaps[12], but have yet
to be realized experimentally due to similar challenges facing quantum confinement. Func-
tionalization through chemical modification[13] or substrate interaction can open gaps by
breaking sublattice symmetry in graphene. However, insufficiently ordered functionalization
or metallic substrates[14] make the transport studies challenging due to reduced mobility
or substrate conduction. These issues are absent in the buffer layer system as the substrate
interaction is known to be ordered and SiC is an insulator. Furthermore, a band gap opened
from substrate interaction does not require precise control of nanoscale features.
1.2 Graphene on SiC
Graphene was first discovered by Van Bommel et al. in 1975[16] by heating SiC until
silicon sublimated from the surface. Once the SiC cooled, the excess carbon rearranged
into layers with a graphitic lattice constant. Forty years later, graphene growth by thermal
decomposition of SiC is one of the most promising growth methods for graphene based
electronics[17]. The reason for this is that current growth techniques[15] allow for precise
layer control and the graphene is well ordered and crystallographically aligned.
There are many methods for growing graphene on SiC. Most studies of graphene growth
have been on the two polar faces, the SiC(0001) Si terminated face (Si-face) and SiC(000¯1)
carbon terminated face(C-face), of hexagonal 4H and 6H SiC[18, 19, 20]. Note that graphene
growth also occurs on other faces and polytypes such as cubic β-SiC(111)[21]. The crystal
structure of 4H-SiC is shown in Fig. 1.2(a). The earliest growth methods consisted of heat-
ing in ultra high vacuum (UHV)[16, 22]. Growth on the C-face and Si-face were found to be
4
8. ab
c
A
B
C
B
a b c
Si-face
C-face
SiC
induction heater
vacuum
leak
graphite
susceptorSi (vapor)
Si-face
C-face
BG
MG
Figure 1.2: Epitaxial graphene on SiC. (a) Crystal structure of 4H-SiC. Si (C) atoms are
yellow (black). The SiC-bilayer stacking sequence is ABCB. The bulk SiC(0001) surface (Si-
face) terminates with a plane of Si atoms, conversely SiC(000¯1) terminates with C atoms
(C-face). (b) Graphene growth on SiC by confinement controlled sublimation [15]. (c)
Schematic of graphene on SiC. Graphene growth is slower on the Si-face compared to the
C-face. The first layer of graphene in the Si-face is called the buffer layer (BG), the second
layer is called monolayer graphene (MG).
quite different. On the C-face growth occurs more quickly and > 30 layers are possible with
many commensurate rotations between the graphene layers[23, 24]. On the Si-face, growth
is slower and only a few layers of graphene form[6]. Furthermore, the graphene layers are ro-
tationally aligned with the SiC to form the so-called (6
√
3×6
√
3)SiCR30◦ reconstruction[20].
The challenge with UHV graphene growth is that large area layer control is not possible
due to high growth rates. Currently, there are a few improved growth techniques such as
growth in an Ar environment[17], chemical vapor deposition[25] and confinement controlled
sublimation (CCS)[15]. In the CCS method, SiC is inductively heated inside a graphite en-
closure [See Fig. 1.2(b)]. The sublimated Si generates a partial pressure due to confinement
within the enclosure. The partial Si pressure is determined by the temperature, crucible ge-
ometry and leak rate. The increase in Si partial pressure slows the graphene growth process
to near equilibrium and causes growth to occur at higher temperatures. For a particular
enclosure, the number of layers grown is determined by the temperature and time. With
the current design, the first layer of graphene, commonly referred to as the buffer layer
(BG), grows at 1400◦C and the second layer (MG) grows at 1550◦C [See Fig. 1.2(c)]. This
5
9. a∗
Gb∗
G
a∗
SiC
b∗
SiC
a b c
Figure 1.3: The (6
√
3×6
√
3)SiCR30◦ reconstruction on SiC(0001). (a) LEED image of a
buffer layer grown by confinement controlled sublimation. (b) reciprocal space interpreta-
tion of (a). The large red (blue) circles are SiC (graphene) rods, smaller red (blue) circles
are (6×6)SiC satellites about SiC (graphene). The graphene rods appear to be commensurate
with the 6
√
3 reciprocal lattice (black dots). (c) Real space 6
√
3 unit cell. The gray (yellow)
circles represent carbon (Si) in the graphene (bulk-terminated SiC). The black diamond is
the 6
√
3 unit cell and the red diamond is the (6×6)SiC quasi-unit cell. The red and blue
filled hexagons emphasize that the (6×6)SiC is not a true unit cell. The red (blue) hexagons
demonstrate the presence (absence) of a graphene carbon atom at the boundaries of the
quasi-unit cell.
convention was adopted since the buffer layer did not possess the electronic properties of
freestanding graphene due to its interaction with SiC. Only when the second graphene layer
forms do angle resolved photoemission spectroscopy (ARPES) measurements observe the
characteristic linear dispersion[19].
1.3 The buffer layer and the 6
√
3 × 6
√
3 R30◦
reconstruc-
tion on SiC(0001)
The history of experimental studies of graphitization on SiC(0001) is characterized by seem-
ingly opposed results and remains to be fully understood. It has been contested whether
the SiC interface is a 6
√
3 or (6×6)SiC reconstruction, if it is bulk-terminated[26], and if the
topmost carbon layer is a full, partial or defected graphene layer[27, 28].
LEED images always suggest the presence of what appears to be a 6
√
3 reconstruction
on SiC(0001) at all stages of graphitization [16, 29, 30]. As a result, layer estimation from
LEED can be challenging and can depend on the growth procedure[29]. The buffer layer
is regarded to form with the initial formation of the 6
√
3 reconstruction. A typical LEED
image of the 6
√
3 is shown in Fig. 1.3(a). The graphene reciprocal lattice vectors are rotated
6
10. 30◦ from SiC and the magnitude is |a∗
g|/a∗
SiC| = 13/6
√
3, i.e. a∗
g = 13/6(a∗
SiC + b∗
SiC). This
leads to a lattice constant of ag = aSiC 6
√
3/13 = 2.462˚A that is only slightly (0.1%)
expanded from graphite (2.460(2)˚A) [31, 32, 33, 34]. Furthermore, a (13 × 13)g graphene
unit cell is commensurate with the (6
√
3×6
√
3)SiCR30◦ unit cell. How a graphene reciprocal
lattice fits onto the 6
√
3 reciprocal lattice is shown in Fig. 1.3(b) and the corresponding real
space unit cell is shown in Fig. 1.3(c).
In constrast to LEED, STM measurements have not produced a satisfying image of a
6
√
3 reconstruction[29]. The primary reconstruction observed is (6×6)SiC [26, 35]. The
difference between the 6
√
3 and (6×6)SiC unit cells are demonstrated in Fig. 1.3(c). The
two measurements are at odds in the sense that an integer multiple of (6×6)SiC reciprocal
lattice vectors cannot describe the graphene position in Fig. 1.3(a). (6×6)SiC features are
observed in LEED images as “satellites,” i.e. there are diffraction rods that surround the
(1 × 1)SiC and (1 × 1)g reciprocal lattice rods that are described by (6×6)SiC reciprocal
lattice vectors[See Fig. 1.3(a) and (b)]. However, it should be noted that (6×6)SiC satellites
surrounding graphene diffraction rods lie on the 6
√
3 reciprocal lattice.
The interpretation of LEED and STM measurements remains an ongoing research
question[36, 37]. One model is that the SiC is unreconstructed. In this model the satellite
rods in LEED are interpretted as resulting from multiple scattering and the (6×6)SiC pattern
seen in STM measurements results from a moir´e pattern between the SiC layer and the
graphene layer. In constrast, the reconstructed model claims that the satellite rods in
LEED are due to structural changes in the interface between graphene and SiC and the (6×
6)SiC seen in STM is imaging the interface reconstruction. Within the reconstructed model,
bulk-terminated[38] and adatom[26] models have been proposed. The bulk-terminated re-
constructed SiC interface model is supported by XPS and ARPES measurements[19] and
the adatom model is supported by STM. What has become clear is that the buffer layer is
a complete graphene layer without substainial defects or incomplete regions[39]. As growth
methods and experimental techniques have improved, the understanding of the buffer layer
electronic structure has changed. Initial ARPES measurements found the buffer layer to be
a wide gap insulator with two significant surface states within the gap making the buffer
7
11. layer unsuitable for electronic applications[19]. Using improved growth methods, this pic-
ture changed and found the buffer layer to be a true semiconductor with no surface states
and a band gap > 0.5 eV [40]. Furthermore, recent structural measurements raise suspicion
that buffer layer may be incommensurate and not a 6
√
3 or (6×6)SiC reconstruction[41] and
that the SiC interface reconstruction may not be bulk-terminated[42].
Theoretical studies of graphene on SiC are limited and remain inconclusive as well.
Depending on the assumptions and calculation method, qualitatively different properties are
predictied. Initial calculations were performed on a smaller, highly stained, bulk-terminated
√
3 ×
√
3 SiC
R30◦ unit cell. While these calculations were on an unrealistic unit cell, they
predicted a wide gap buffer with a metallic state from unbonded Si[43, 44] that seemingly
agreed with initial APRES measurements[19]. On the other hand, calculations of the full
bulk-terminated 6
√
3 buffer vary in predictions from metallic[45], to supporting the insulator
picture of the (
√
3×
√
3)SiC model[46], or remaining silent on its electronic properties[47, 48].
Furthermore, it is not clear why the bulk-terminated reconstruction should be 6
√
3 as there
are other more energetically favorable interface structures, such as (4 × 4)SiCR24.2◦[48].
From all the theoretical studies conducted, none predict the most recent electronic structure
measurements of an improved buffer layer or STS measurements with a gap ∼1 eV[39, 26].
It is clear that more detailed structural measurements are needed to guide further theoretical
study and clarify previous experimental results.
8
12. CHAPTER II
PRELIMINARY RESULTS
The initial work for this thesis proposal attempts to resolve the contradiction between the
theoretical and experimental buffer layer band structure. Using improved growth methods,
the first high resolution surface x-ray diffraction measurements of the buffer layer system
were obtained. These results show that contrary to the last forty years, the buffer layer
system is not the assumed (6
√
3×6
√
3)SiCR30◦ reconstruction. Rather, the graphene lattice
is incommensurate with bulk SiC and engages in a mutual modulation with the SiC interface.
With this new structural model, the measured band structure can be described with a simple
tight binding model.
2.1 The incommensurate SiC(0001) interface
In the traditional buffer layer picture, the commensurate 6
√
3 structure gives rise to 6th
order diffraction rods around the bulk SiC reciprocal lattice rods [see Fig. 1.3 and the insert
in Fig. 2.1(a)]. However, high resolution SXRD measurements reveal that the satellite rods
are symmetrically shifted away from the commensurate 6th order positions and towards the
bulk SiC rods. The incommensurate rods along k in Fig. 2.1(a) are ±q1 = K − GSiC
0,1 whose
magnitude is q1 = qoa∗
SiC, where qo = 1/6(1 + δ) and δ = 0.037(2).
This behavior is a classical result of an incommensurate system[49]. The contracted
satellite positions are a direct result of the commensurate unit cell positions, R, being
modulated by a function, η(R, q). The modulation can be Fourier expanded so that the
new modulated positions, r, are given by[50],
r = R +
d
j=1
ηj sin (qj · R + φj). (2.1)
Each Fourier component has a corresponding amplitude ηj, wavevector qj, and phase φj.
The number of Fourier components is d. From this description the diffraction condition and
9
13. Figure 2.1: Diffraction results from the incommensurate graphene-SiC(0001) systems. (a)
SXRD radial k scans, (0, k, 0.1), around the SiC (0, 1, l) rod (see schematic in the insert).
The background-subtracted intensity is instrument corrected. Data is for the BGo (blue)
and MG (grey) films. Dashed lines mark the commensurate 5/6th and 7/6thpositions in
reciprocal space (black circles in insert). The buffer satellite rods are contracted relative to
the commensurate positions towards the (0, 1, l) rod while the monolayer rods are nearly
commensurate. (b) Radial scan through the nominal graphene (0, 3, 0.1)G rod for the BGo
(blue ◦) and MG (grey ◦) films. Dashed line marks the expected position for a commensurate
6
√
3 graphene film. Blue arrow shows the expected (0, 3, l)G position from Eq. 2.7. The
monolayer film has a contribution from the MG (red line) and the BGML rods (black line).
The green (red) arrow marks the position for graphite (theoretical graphene). The arrows’
horizontal bar represent their known uncertainties. (c) Radial width of graphene rods as a
function of K for BGo (blue ◦), MG (red ◦), and BGML (grey ◦).
10
14. intensity can be calculated. In the kinematic approximation the scattering amplitude is a
Fourier transfrom of the electron density, which is modeled as a delta function distribution,
A(K) =
r
freiK·r
, (2.2)
where fr is the atomic form factor of the atom at position r. Substituting eq. 2.1 into eq. 2.2
and using the Jacobi-Anger expansion eiZ sin θ = ∞
p=−∞ eipθJp(Z), where Jp(Z) is a Bessel
function of the first kind, gives,
A(K) = F(K)
R
∞
{pj}=−∞
ei(K+ j pjqj)·R
ei j pjφj
d
j=1
Jpj (K · ηj). (2.3)
F(K) = Rb
freiK·Rb is the average structure factor from the basis atoms, Rb and is as-
sumed to be slowly varying in subsequent calculations. The integrated intensity is calculated
from the scattering amplitude by I = dK3|A(K)|2,
I(K)
N2|F(K)|2
=
{pj}{pj }
d
j,j
Jpj (K · ηj)Jpj
(K · ηj )ei(pjφj−pj φj )
. (2.4)
Equation 2.3 describes a set of satellite rods through the diffraction condition,
K = G −
d
j
pjqj, (2.5)
where G is a reciprocal lattice vector of the SiC (G(SiC)) or graphene G(g) unmodulated lat-
tice, pj is an integer for the corresponding qj and {pj}{pj } in eq. 2.4 impose a restriction
of the sums such that,
d
j
pjqj =
d
j
pj qj . (2.6)
What is unique about the buffer system is that the buffer layer and the SiC interface layer
are mutually modulated. This is known from the fact that the spacing between the buffer
G
(g)
1,1 and the SiC G
(SiC)
0,2 is an incommensurate wavevector, i.e. q1 = G
(SiC)
0,2 − G
(g)
1,1. This
behavior occurs when the buffer graphene modulation can be Fourier expanded in SiC
reciprocal lattice vectors and vice versa. The allowed incommensurate wavevectors can be
generalized as,
11
15. {q} = G(SiC)
− G(g)
. (2.7)
Since the graphene lattice constant is incommensurate with the 6
√
3 [see Fig. 2.1(b)], so
is the period of the modulation. The mutual modulation is further confirmed by the excellent
agreement between the measured and expected (G
(g)
0,3 = G
(SiC)
1,1 −q1+q2) position of the buffer
G
(g)
0,3. Note that the incommensurate buffer lattice constant is larger (2.469(3)˚A) than the
expected 6
√
3 value (2.462˚A, vertical dashed line) due to the shift to lower K||. Furthermore,
the lattice constant is larger than graphite (2.460(2)˚A) and theoretically isolated graphene
(2.455(3)˚A)[51, 52, 53].
An increased lattice constant is consistent with stronger interlayer interaction in layered
materials[54] and with the view that some of the carbon atoms bond lengths increase due to
some degree of hybridization with Si in the SiC interface. Also, the incommensurate lattice
constant is consistent with previous measurements[41] although they incorrectly attributed
the expanded lattice constant to a buckled buffer graphene layer to claim the buffer layer
was commensurate with a 6
√
3 unit cell. Rather, in the context of mutual modulation the
buffer in-plane lattice constant is properly understood as being truly incommensurate.
The modulation amplitude in the SiC interface layer and the buffer graphene layer can
be quantified by comparing the measured, instrument corrected[55], integrated intensity of
the incommensurate satellite rods to the calculated intensity. The intensity was numerically
calculated with a large number of lattice points by inserting eq. 2.1 into 2.3. To obtain
a meaningful estimate of the modulation, the number of terms in η(R, q) must be limited.
Experimentally, the satellite rods of significant intensity were first order. The minimum
number of wavevectors needed to reproduce the symmetry of the satellites was found to
be d = 3. The three {q} are shown in Fig. 2.2(a). They are of equal magnitude and
directed along q1 = −qob∗
SiC, q2 = qoa∗
SiC, and q3 = qo(b∗
SiC − a∗
SiC). Note that from the
non-orthongonality of {q}, there are multiple sets of {p} for a given satellite rod that
contribute to the diffraction intensity. For example, −q1 = q2+q3 and satisfies the restriction
{pj}{pj } . From symmetry considerations, the modulation amplitudes, {η}, are assumed
to have the same magnitude, η(SiC) or η(g), and parallel to their respective {q}. Multiple
12
16. Figure 2.2: SXRD derived structure of the buffer-SiC interface. (a) The instrument cor-
rected integrated intensity of the satellite rods around the (01l) rod. Crosses mark the
commensurate 6th order rods. The arrows show the three IC wavevectors. The gold circle’s
area are proportional to the measured intensity of the satellite rods. The red circle’s area
are proportional to the fit intensity described in the text for η(SiC) =0.11aSiC. (b) The η
(SiC)
||
dependence of the calculated intensity for the satellite rods shown in (a). The intensity is
normalized to N2. The vertical blue box shows the range of η
(SiC)
|| that best fit the measured
values of all seven rods. The circles represent the normalized experimental intensity values
for the satellite rods.
orientations for the modulation amplitudes were tested and these choices were found to
provide the best fit. Further, the intensity was found to be insensitive to the choice of {φ}.
Figure 2.2(b) shows how the calculated intensity of the satellite rods around G
(SiC)
0,1 vary
as a function of η(SiC). The intensity was normalized to N2, where N is the number of
lattice points in the numerical calculation. When η(SiC) → 0, I → 0 for all satellite rods
and I → 1 at K = G. As η(SiC) increases, the intensity is not symmetric in η(SiC) about zero
and the satellite intensities develop an asymmetry like the observed experimental pattern
[see Fig. 2.2(a)]. The range of η(SiC) with the proper intensity pattern is highlighted by the
blue box in Figure 2.2(b). The best fit value is η(SiC) =0.11(4)aSiC. The same analysis was
performed on the graphene satellites and found a weak, but non-zero, in-plane modulation
of η(g) < .01aSiC.
The incommensurate modulation can be visuallized by plotting the relative density
change of the SiC interface layer. By considering eq. 2.1 as a coordinate transformation, the
normalized density change relative to the unmodulated system is given by ∆ρ/ρ = J−1 −1,
where J−1 is the inverse of the Jacobian determinant for the transformation of R → r. The
13
17. Figure 2.3: Relative density ∆ρ(x, y)/ρ map of the incommensurate SiC interface using the
measured {q} and best fit η
(SiC)
|| . The grey circles and hexagonal mesh overlay represents
interface Si and graphene, respectively. The commensurate 6
√
3 unit cell is marked in red.
Black arrows show the three incommensurate wavevectors.
colormap in Fig. 2.3 shows that the SiC interface consists of a super-hexagonal network with
a period of λ = 6(1 + δ)aSiC. The boundaries have a higher density than bulk terminated
SiC. Note that while the density modulation is periodic, the atomic positions of both the
SiC interface and the buffer graphene are not periodic. The network is very similar to STM
images of the buffer layer[29, 27, 35, 26].
The exact structure of the SiC interface and the driving force for the incommensurate
phase remains to be determined. It is unlikely that a simple sinusoidal modulation used
to fit the data is the complete picture. Recent work by Emery et al. [42] may provide a
clue. They show that the interface layer below the buffer graphene layer has a lower Si
and higher C concentrations than bulk SiC. Silicon vacancies and/or substitutional carbon
could give rise to different bonding geometries that could produce strains sufficient to drive
the incommensurate modulation. On the other hand, a comparable strength in-plane and
interlayer interaction coupled with subsequent changes in bond length may be sufficient to
drive the incommensuration in a bulk-terminated system.
14
18. 2.2 Semiconducting graphene from interface interaction
Although the exact structure is still unknown, the discovery of an incommensurate mutual
modulation in the buffer-SiC system allows the electronic structure of the buffer layer to
be studied and explained in a new context. As will be shown, a simple tight binding
model can reproduce ab initio calculations and experimental ARPES measurements. First
a description of the tight binding calculations is provided followed by a description of the
model and parameters used to study the buffer-SiC interaction.
Given that η(SiC) η(g) and η(g) is small, the buffer-SiC system is modeled as an
unmodulated graphene layer above a modulated triangular lattice of Si atoms at the SiC
surface for all calculations. The Si lattice is rotated 30◦ from the graphene lattice vectors.
Tight binding calculations require periodicy. As a starting point, a (13×13)g graphene unit
cell is commensurate with a 6
√
3 SiC unit cell. In this model, only the influence of Si on the
buffer graphene is studied. A Si interface atom bonds to its nearest neighbor graphene atom
if the in-plane Si-C distance is within a maximum cutoff radius, rcut. Bonding is modeled
through an onsite potential to the carbon atom in the graphene layer. Using a single orbital
nearest neighbor tight binding model of the graphene π-electron network, the Hamiltonian
reads,
H = −t
R,R i,j rSi
(a†
R+ri
aR +rj
+ V θ (rcut − rSi) a†
R+ri
aR+ri
+ H.c.), (2.8)
where R (R ) specifies the unit cell, r specify the C atom positions within the graphene
unit cell, i, j represent that R + ri and R + rj must be nearest neighbors to the carbon
atom and rSi represents the in-plane nearest neighbor distance for a Si atom in the interface
layer to the graphene C at ri. t ≈ 2.8 eV is the transfer integral between nearest neighbor
C sites and a†
r (ar) are operators that create (annihilate) a carbon π-electron at r. V is
the onsite potential due to bonding from with a Si interface atom. θ (rcut − rSi) is the
Heavyside step function that is 1 if rSi < rcut and 0 otherwise, where rSi is the nearest
neighbor distance of a Si interface atom to a C graphene atom. The Fourier transformation,
ar = 1/
√
N k
eik·rφk
(r) is applied to the Hamiltonian, where φk
(r) is the operator that
annihilates a pz orbital at r. After this transformation, the sums over R and R are identical
15
19. and the Hamiltonian becomes,
H = −t
i,j
eik·ri,j
φ†
k
(ri) φk
(rj) + V θ (rcut − rSi) φ†
k
(ri) φk
(ri) + H.c., (2.9)
where ri,j = (R + ri) − (R + rj). The solution to the Schr¨odinger equation is Ψn(k) =
r Cn
r φ†
k
(r) where the coefficients are obtained from the solution to the eigenvalue prob-
lem, det H(k) − ES(k) = 0. The overlap of the pz orbitals is assumed to be negligible.
Therefore, the elements Sm,n = δm,n. The matrix elements of the Hamiltonian are
Hm,n = −t eik·rm,n
δrm+rm,n,rn + V θ(rcut − rSi)δrm,rn . (2.10)
The method used to compare the calculated band structure with the ARPES measure-
ments in Fig. 2.4 is to project the band structure from the supercell Brillouin Zone (BZ)
to the larger primitive graphene BZ[56]. When unfolding the band structure, a weight is
assigned to each eigenvalue at each k value;
Wn(k) =
1
n
rA,i
Cn
rA,i
(k)
∗
rA,i
Cn
rA,i
(k) +
rB,i
Cn
rB,i
(k)
∗
rB,i
Cn
rB,i
(k)
. (2.11)
Here rA/B refer to C atoms in the larger unit cell that unfold to the A(B) sublattice.
Neglecting effects such as polarization, photoelectron diffraction, etc., the ARPES intensity
can then be approximated as;
I(E, k) =
n
i
Wi(k)δ(E − En(k))dE. (2.12)
Assuming the pz orbitals are sufficiently localized, the wavefunction coefficients can also
be used to model an STM measurement. This is done by calculating the charge density
[See Fig. 2.4(c)] for a given energy[10]
ρ(r, E) =
n,k
|Cn
r (k)|2
δ(E − En(K))dE. (2.13)
Both the ARPES intensity and charge density were broadened in energy by ±0.1 eV to
better represent the experimental measurements.
16
20. Figure 2.4: Comparison of the theoretical the incommensurate graphene band structure
with experimental ARPES data. (a) The commensurate 6
√
3 buffer structure derived from
ab initio calculations in Ref. [45]. Black circles are carbon unbonded to the SiC. Gold circles
are carbon bonded to Si in the interface layer below. The NC regions (blue hexagons) and
the carbon chains are marked. (b) A model structure based on modulated SiC layer using
the experimental value, η(SiC) = 0.11aSiC (same color scheme as (a)). Red dashed hexagon
marks the boundary of an isolated graphene island. (c) The calculated charge density
(arbitrary units) at E =−0.6 eV for the structure in (b). (d) TB bands (red) mapped onto
the graphene BZ from the commensurate structure in (a). The low energy bands from the
ab initio commensurate structure are overlaid (black dashed line). (e) DOS for the TB
bands in (d). (f) TB calculated bands (red) from the modulated structure in (b). The
negative 2nd derivative of the experimental ARPES bands (blue) are overlaid. The π-bands
from a 2% monolayer have been subtracted from the experimental bands. (g) DOS for the
TB bands in (f). The direct 0.8eV bandgap is marked.
17
21. rSi
distance (ag
)
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Counts(nSi
)
0
5
10
15
20
25
rcut
1/2acc
a b
Figure 2.5: Bonding geometry from ab initio calculation[45]. (a) shows the bonding con-
figuration of the 6
√
3 × 6
√
3R30◦
SiC or (13 × 13)g unit cell do to an assumed unmodulated
bulk-terminated SiC(0001) interface. The graphene lattice vectors are the black arrows,
the SiC interface lattice vectors are the green arrows. The black (gray) circles are carbon
atoms in the graphene layer bonded (did not bond) to Si. The large (small) yellow circles
are Si atoms bonded (not bonded) to C in the graphene layer. The blue circles around a
pair of C atoms are the same distance from the nearest Si within rcut. The red lines indicate
situations where bonding within rcut did not occur. (b) Histogram counting the number of
Si atoms with a nearest neighbor planar distances between Si in the interface layer and C in
the graphene layer, rSi. The yellow (red) histogram of rSi is for bonded (not bonded) Si to
graphene C atoms. The most notable feature is the overlap near .34ag, which is highlighted
in (a).
In order to properly use this tight binding model, the parameters, V and rcut must be
obtained. To do this, the parameters were determined so that the band structure from the
tight binding calculations reproduced previous ab initio calculations[45]. In their calcula-
tions, a complete graphene layer commensurate with the 6
√
3 rested on a bulk-terminated
SiC interface[45] and was then allowed to relax. Figure 2.4(a) shows a visualization of the
predicted bonding configuration. Not all Si interface atoms bonded the graphene C atoms.
They found that 79% of the interface Si bonded to 25% of the BGo graphene C atoms. The
bonding pattern is divided into two regions: a nearly commensurate (NC) region and the
boundaries between the NC regions. In the NC region, the bonding and atomic positions
are close to previous calculations[43] on a (
√
3 ×
√
3)SiC where the unbonded C reseme-
bles benzene-like rings. At the NC boundaries, incomplete hexagonal chains form and are
responsible for the low energy electronic metallic band structure.
18
22. An analysis of the bonding configuration is presented in Figure 2.5(a). The histogram
in Fig. 2.5(b) shows that when rSi > 0.35ag, the Si atom will not bond. The maximum
bond length does not have a sharp cutoff, i.e. there are a few cases where rSi = 0.35ag
and bonding does and does not occur. These four cases are highlighted by red lines in
Fig. 2.5(a). An additional complication arises when 1/2acc < rSi < 0.35ag (acc is the C-C
bond length in buffer graphene layer). In this case 2 C atoms may be equidistant from
the nearest Si atom. In the ab initio calculations, only one C atom bonds to the interface
Si atom[See the blue circles in Fig. 2.5(a)]. Using this information, a simplifying choice
rcut = 1/2acc is used for the incommensurate system.
The onsite potential, V , was chosen by comparing the TB results to the bands from
the ab initio calculations. This was done by adopting the predicted bonding shown in Fig.
2.4(a). A constant V was assigned to those C atoms in the graphene layer that were bonded
to Si in the SiC surface. A large range of V were tested and the resulting band structure
using those V ’s was compared with the ab initio band structure. Perhaps surprisingly, it
was found that V ≈ (−4±2)t in the TB calculation best reproduced the ab intio bands[See
Fig. 2.4(d)]. The large range in V indicates that the tight binding model is robust and
that the key physics from the ab initio calculations are being represented. The agreement
and simplicity of this model immediately provides insight into the ab initio calculations. It
predicts the interaction between SiC and graphene should be strong, but at the same time
the π-bonds are preserved. Note competing interactions of similar strength is a tell-tale
sign of a possible incommensurate phase. Also, it predicts that effects such as strain in the
buffer graphene layer are at least second order to the Si-graphene interaction in describing
the band structure. While both calculations are in excellent agreement, they clearly do not
predict the ARPES semiconducting experimental bands plotted in Fig. 2.4(f).
The effect of modulation must be considered to describe the experimental band struc-
ture within this tight binding model. The bonding configuration and band structure was
calculated for many values of η(SiC). As may be expected, significant changes in bonding
configuration occur when the Si interface atoms are modulated according to eq. 2.1. As
such, significant changes occur in the band structure due to the strong coupling through
19
23. h(SiC)
=0.09aSiC
h(SiC)
=0.052aSiC
1
0
-1
-2
-3
E(eV)
Г K M
1
0
-1
-2
-3
E(eV)
Г K M
a b
c d
hSiC
(aSiC
)
0.1 0.2 0.3
BandGap(eV)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
e
Figure 2.6: Bonding and band structure dependence on η(SiC). (a) and (b) The large
(small) circles indicate unbonded (bonded) graphene to the Si below. As the modulation in
the SiC interface increases the bonding configuration changes. At η(SiC) = 0.052aSiC shown
in (a) the chain boundary of the unmodulated case broadened and opened a band gap of
0.26 eV shown in (c). At η(SiC) = 0.09aSiC shown in (b) the bonding becomes more like
graphene islands and a larger gap forms as shown in (d). (e) The calculated band gap as
a function of η(SiC). As η(SiC) increases, the graphene “island” develops and the band gap
increases to a value that appears to saturate. Dashed line shows the average value of η(SiC)
that produces a given gap.
20
24. the onsite potential. Figure 2.4(b) shows the bonding configuration using the experimental
η(SiC). The modulation descreases bonding to the graphene layer by nearly 40% compared
to the unmodulated case in Fig. 2.4(a). Half of the NC regions in the unmodulated struc-
ture convert into a large region of unpeturbed graphene “islands” that correspond to low
density regions in the SiC interface layer shown in Fig. 2.3. The graphene between the
islands, aligned with the high density boundaries, have a higher number of bonds to the
interface Si as might be expected.
The calculated semiconducting bands look remarkably similar to the measured ARPES
bands for the experimental η(SiC) = 0.11aSiC[See Fig. 2.4(f)]. The predicted band gap
is 0.8 eV as shown in the density of states (DOS) in Fig. 2.4(g) and is consistent with
previous STS measurements[39, 26]. Also, the predicted charge density from the three
highest occupied bands show weak localization at the island edges [See Fig. 2.4(c)] and give
rise to a charge density that strongly resembles previous STM measurements[29].
When studying a large range of η(SiC), it was found that graphene island formation and
subsequent band gap opening was a robust feature of the modulated system. Figure 2.6
(a)-(d) shows the bonding configuration and band structure for two additional values of
η(SiC) other than the experimental value. When η(SiC) < 0.1aSiC, the structure and band
gap are sensistive to small changes in η(SiC)[See Fig. 2.6(f)]. However, even in this range
graphene island formation is sufficient to open band gaps > 0.25 eV . When η(SiC) exceeds
0.1aSiC, a graphene island is fully formed and no significant change in bonding occurs up
to 0.36aSiC. In this range the band gap is largest and nearly constant.
The fact that the unbonded graphene “island” configuration is present for such a large
range of η(SiC)’s and that the opening of a bandgap is prevalent over this same range
lends a great deal of weight to the idea that the incommensurate system is responsible
for the buffer’s semiconducting properties. Coupling this to the fact that the calculated
band structure, using the experimental value of the modulation, reproduces the important
features of the experimentally measured bands supports the importance of the modulated
interface controlling the buffer’s electronic properties.
21
25. Figure 2.7: The effect of ML graphene growth on the buffer band structure. (a) ARPES
bands at the BGo layer K point (kx is perpendicular to ΓK, hν = 70 eV). A Dirac cone
from a 2%ML graphene layer is also visible. (b) A negative 2nd derivative filter of the BGo
bands in (a). (c) A similar 2nd derivative filter for a MG film. Red dashed lines mark the
approximate 0.4 eV shift of the buffer bands.
2.3 Buffer layer stability
Finally, the stability of the buffer layer is addressed. When a monolayer (MG) of graphene
forms above the buffer layer, there are changes in the buffer’s structure. The buffer with
MG on top is distinguished as BGML from the bare buffer layer BGo. Once the MG forms,
the satellite positions and the lattice constant become nearly commensurate (δ < 0.02) with
the bulk SiC [See Fig. 2.1(a) and (b)]. The MG lattice contracts relative to BGML making
the MG incommensurate with both the BGML and SiC. The lattice constants for the buffer
and MG systems are summarized in Table 2.1. Note that the MG lattice constant is nearly
that of theoretically isolated graphene and contracted from graphite. The contraction from
graphite is due to comparitively reduced interlayer interaction. Also, the MG interacts
with only one layer and the incommensuration reduces the interlayer coupling compared to
Bernal stacking. This contraction is analogous to non-Bernal stacked graphene layers on
C-face SiC [See Table 2.1].
3
This work
5
From Ref. [23].
4
Similar values were measured by Schumann et al., [41].
1
From Ref. [51, 52, 53].
2
From Refs. [31, 32, 34, 33].
22
26. Table 2.1: Comparisons of graphene lattice constants, their relative strain (∆a) compared
to theoretical graphene, RMS strain rms, and long range order
Graphene Form Lattice constant (˚A) ∆a (%) rms (%) Order (nm)
Theoretical MG 2.453(4)1 - - -
Graphite 2.460(2)2 +0.28 - -
BGo 2.469(3)3,4 +0.70 0.2 60
BGML 2.462(3)1,3 +0.40 0.6 43
MG 2.455(3)1,3 +0.10 0.3 43
C-Face multilayer 2.452(3)5 -0.04 - 300
There are two additional changes when the MG forms. First, the system becomes more
disordered (30% decrease in long range order) as evidenced by the broader satellite rods in
Fig. 2.1(a). Also, the BGML develops a large RMS strain, rms. RMS strain presents itself
as K-dependent broadening ( rms ≈ ∆K/K). The plot of ∆K vs. K in Fig. 2.1(c) shows
that BGML has the largest slope, i.e. the largest rms. The RMS strain in MG is smaller,
presumably due to strain relaxation allowed by weaker coupling to BGML. However, BGo
presents the lowest overall RMS strain.
It was assumed that the strong buffer-SiC interaction meant the buffer band structure
did not change significantly once the MG formed. Now that a structure change in the buffer
was demonstrated in Fig. 2.1 upon MG formation, it is prudent to revisit how or if the
BGML differs from BGo. Figure 2.7(a) shows the ARPES spectra from the BGo layer. The
π-bands are broad (∆k ∼ 0.35 ˚A
−1
) consistent with q ∼ 0.38 ˚A
−1
. In order to compare
the BGML bands with the BGo, we have plotted a 2nd derivative spectra of the buffer and
MG bands in Fig. 2.7(b) and (c). This compensates for both the ∆k broadening and the
photoelectron attenuations through the MG.
Figure 2.7(c) shows that the semiconducting π-bands are still present with the MG
above. Although the BGML bands intensity is weak, it is consistent with a complete buffer
layer after correcting for attenuation. There is, however a change in the BGML bands
compared to the BGo bands. The π-bands are pushed to lower binding energy by ∼0.4eV
compared to the BGo bands and the band near EF appears to have less dispersion than
the BGo case. While there is a small energy gap between the BGML layer bands and
EF , the experimental error could also support the BGML layer being metallic. Note that
23
27. η(SiC) < 0.05 aSiC (the uncertainty is due to the increased disorder in the BGML). The
low value of η(SiC) is consistent with a buffer layer structure closer to the commensurate
structure that would give rise to either a small gap or metallic bands. The fact that MG
is incommensurate with BGML provides new insight into why graphene grown on the Si-
face has historically lower mobilities than C-face graphene[4, 57, 58]. The incommensuration
may give rise to a quasi-random network of MG-BGML coupling that act to increase random
scattering and thus lower the mobility.
24
28. CHAPTER III
FUTURE WORK
The discovery of incommensurate mutual modulation in the buffer-SiC system has disrupted
the past forty years of understanding of graphene growth on SiC(0001). Its importance is
highlighted by its role in forming semiconducting graphene and reconciling experimental
and theorectical studies. Furthermore, finding semiconducting graphene in the buffer layer
comes at an appropriate time where many research programs have moved on to other less
favorable 2D materials that intrinsically possess a band gap. Graphene on SiC is an ideal
platform for graphene electronics and the semiconducting buffer provides a crucial missing
element. While significant progress has been made towards understanding and producing
the buffer layer, there are many exciting avenues for further exploration that will improve
our understanding of 2D materials and propel graphene from pure research to applications.
The origin of the incommensurate phase in BGo and why BGML becomes nearly com-
mensurate is unknown and can be explored further by detailed experimental studies. It is
not conclusive from initial SXRD measurements that the SiC interface is a bulk-terminated
reconstruction or one with adatoms, vacancies, or varied atomic concentrations. One tech-
nique available to address this questions is x-ray standing wave enhanced x-ray photoelec-
tron spectroscopy (XSW-XPS) that enables a layer by layer estimate of atomic concentra-
tions. XSW-XPS measurements have already been performed on UHV grown multilayer
graphene samples[42]. However, our measurements show that there is a structural change
when a monolayer forms above the buffer layer. As such, XSW-XPS measurements on a
bare buffer layer and monolayer using improved growth methods will provide insights into
the origin of the structural changes and the nature of the incommensurate phase.
Further detailed SXRD measurements of the buffer layer system can also give insight
into the incommensurate structure. Initial measurements focused on the in-plane modula-
tion period and amplitude. However, since the mutual modulation consists of at least two
25
29. layers, an investigation of out of plane modulation is needed and can be accomplished by l
scans of graphene, SiC and modulation related diffraction rods. Additionally, incommensu-
rate phases are known to have temperature dependence[49]. Characterizing the temperature
dependence through SXRD, ARPES and Raman measurements will provide insights into
the type of incommensurate system and the existence of different phases. STM measure-
ments will complement these studies and help take SXRD measurements beyond 1st order
estimates of density changes. Furthermore, recent STM measurements found substaintial
coverage of small particles above the buffer layer. What these particles are and how to
remove them will be critical towards producing a clean buffer layer for use in electronic de-
vices. The most recent set of SXRD measurements will assist in identifying these particles
by extracting the various bond lengths present in these particles. Chemical identification
will provide assistance in the proper surface treatment required to remove them.
A proper understanding of the buffer-SiC mutual modulation requires theoretical studies
to be carried out in tandem. The current tight binding study shows that SiC modulation can
induce band baps in graphene that are consistent with experimental ARPES bands. Also,
the comparable interaction strengths of graphene in-plane and interface interaction suggest
the possibility of an incommensurate phase. However, the buffer layer still requires more rig-
orous study through ab initio calculations. Currently, only relaxation of a bulk-terminated
SiC interface has been studied, but the SXRD measurements show that relaxation from a
modulated configuration should be considered as well. Unfortunately, ab initio calculations
alone may not be sufficient to understand the incommensuration. Most studies require the
imposition of some periodicity. Tight binding allows for the study of larger periodicities
more similar to the incommensurate structure. Therefore, a refinement of the tight binding
model from ab initio calculations is meritted in order to assist in studying the origin of the
incommensurate phase.
The CCS growth method provides substantial improvement in order and layer uniformity
over UHV growth. However, in the CCS design, the temperature and Si partial pressure are
linked by the crucible design. Developing new growth techniques that allow independent
exploration of temperature and Si partial pressure will allow for optimization of sample
26
30. order, the study of all buffer-SiC phases and potentially a platform for bandgap engineering.
Ultimately, the excitement surrounding the buffer layer is its potential for use in elec-
tronic devices. Key questions remain as to how to contact, gate, or dope the buffer layer and
if these processes changes its properties. For example, it was shown that the incommensu-
rate modulation results from a mutual interaction between graphene and the SiC interface.
It is unknown how sensitive the buffer may be to external influences such as the deposition
of a gate dielectric. The interaction with a gate or contact material may be strong as the
SiC interaction seems to partially sp3 hybridize the buffer. This may cause the buffer to
be more reactive and potential changes in the electronic structure of the buffer layer will
require carefull study. These effects can be characterized through SXRD measurements of a
buffer layer with a gate material deposited on top as well as ARPES if the photon energy is
high enough to penerate the gate or contact material. Whether or not buffer can be doped
can characterized by ARPES after sequential submonolayer depositions of K or Cs atoms
in UHV.
27
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